According to the specifications for a Gateway SX-2370 (as posted at the top of this page next to your question) the desktop has an integrated video adapter. (This means that it is hard mounted onto the motherboard and usually cannot be replaced easily).
It also has both HDMI and vga video outputs. Have you tried both to see if they are both faulty or if it is only one type of video output that is faulty? (I also assume that you tested the monitor using the same input signal type and port i.e. HDMI or vga, as what you were connected to from the desktop).
If as @jimfixer suggests that you checked for "beeps" on start up and there are none it possibly means that the graphics adapter whilst it is is recognized by the system doesn't have an output.
If this is the case you would have to change the motherboard as again, according to the specifications, there aren't any PCI, PCIe or AGP slots available to insert a new graphics adapter.